The Redundancy of Proof for the Pythagorean Theorem by Maclain Hunter

Authored by Maclain Hunter · first-party-authored · authored-foundational-record

Stored 4P source diagnostic: unresolved · finding unverified_concern · admission admitted_provisionally. This intake result is separate from the theorem checks of the exact propositional sequent.

The Redundancy of Proof for the Pythagorean Theorem by Maclain Hunter: Premises: Let P represent Logic and the Logical Order inherent in deriving trigonometry (a) from the Pythagorean Theorem (b), deriving the Pythagorean Theorem from syllogistic logical reasoning (lower order formal logic), and then deriving lower order syllogistic logic (c) from the Highest Order of Logic (Universal Logic) also called the Logos (d) or “Universal Logos” (containing all logoi and finite reasoning). Let Q represent proof, as well as the contradiction inherent in the assumption that the validity and real-world success of trigonometry is not demonstration of the validity of the Pythagorean Theorem. Let R represent the statement “the Pythagorean Theorem is valid and true, as demonstrated by its continuous practical and accurate function |= the Pythagorean Theorem is valid and true, as demonstrated by its continuous practical and accurate function”. Let R also represent the contradiction that arises from assuming that the Pythagorean Theorem is not inherently “proven” (aligned with Logos demonstrably) by being a valid geometric application of algebra in syllogistic form. Assuming the negation of the set of premises {P, Q, R} leads to logical contradiction, therefore the set of premises is true. Thus the reasoning holds that: {P, P <-> Q, Q <-> R} |= (P & Q) & R Conclusion: {P, Q, R}: The Pythagorean Theorem is already self evident and is proven by its own valid and true identity, which is what any proof or logics must achieve (in any case) - so they are objectively consistent with Logic being Logic. It is absurd to think the Pythagorean Theorem lacks Coherence (Logic) and thus “lacks proof”. It is self-evident in the same way a syllogism with true premises is. The Pythagorean Theorem can, in fact, only leas to a “wrong answer” when the values that are used are incorrect and don’t actually match the triangle at hand - which is not a flaw in the theorem but in the use of it. The Logical Order of concepts a, b, c, and d, representing trigonometry, the Pythagorean Theorem, syllogisms, and Logic (Universal Logos) is as follows: {a, a <-> b, b <-> c, c <-> d} |= (d -> c) -> (b -> a) Truth Table Analysis: valid ↓ ↓ ↓ ↓ PQR⎪P P↔Q Q↔R╞(P&Q)&R ―――――――――――――――――――――― TTT⎪T TTT TTT TTT TT TTF⎪T TTT TFF TTT FF TFT⎪T TFF FFT TFF FT TFF⎪T TFF FTF TFF FF FTT⎪F FFT TTT FFT FT FTF⎪F FFT TFF FFT FF FFT⎪F FTF FFT FFF FT FFF⎪F FTF FTF FFF FF Additional Note: If you don’t accept that the algebra of the Pythagorean Theorem is proven to not be invalid and arbitrary by its proper description of concepts in geometry, that have been shown to align, then identifying “proof” (alignment with Universal Logos) is surely not your forte. It is like thinking “2+2=4” needs external validation, or that the Law of Identity (which is what it is consistent with itself) needs external law. It’s not so. Like the syllogism is a universally valid form of proving C based on A and B the Pythagorean Theorem is a more specific application of this same kind of Logic, tailored to the problem of figuring out a tangent based on sine and cosine, or any combination of those, using two (out of the sine, cosine, and tangent) to prove the remaining/unknown one. Truly, to deny the Pythagorean Theorem which follows and gives example of valid syllogistic form would be to defy and undermine Logic in perhaps its most classical and sound form. Thus the Pythagorean Theorem can also be proven through contradiction by assuming that it is false and not possible to logically proof, and by inspecting the absurdity and negative implications for other aspects of Logic (like syllogistic form) that then arise - consistent with that line of reasoning. It would make sense if Pythagoras had understood this and that it is why arbitrarily “external” proof of the theorem was not such a priority for him. Q.E.D.

Lean 4 verification and timestamps

The machine formalization date is separate from the original authored work. See the verification record for exact checked scope and any unresolved parts.

Logic-centered continuity

This authored proof is connected to Logic, Logos, God, and the self-evident Ground of Reality, and to the distinct systems 4P Engine, MCB Core, 4P-MCB, and the source-first Canon. Formal proof, empirical evidence, and system-generated records remain separately typed.